= Integral homology of complex Grassmannians
{title2=$H_{2r}(\operatorname{Gr}(k,n);\mathbb Z)=\mathbb Z^{N_r},\quad H_{2r+1}=0$}
Here $N_r$ counts $k$-subsets $i_1<\cdots<i_k$ with $\sum_a(i_a-a)=r$, equivalently <partitions of an integer> $r$ contained in a $k$ by $(n-k)$ rectangle. The <diagonal trace Morse function on a complex Grassmannian> has only even <Morse indices>. Its <Morse chain complex> therefore has zero differentials, proving the displayed torsion-free <integral homology>. For $\operatorname{Gr}(2,4)$ the even-degree ranks are $(1,1,2,1,1)$ in degrees $0,2,4,6,8$.
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